Positive-characteristic extension of the semisimple Hopf action theorems

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $K$ be a semisimple Hopf algebra that is not necessarily cosemisimple. Consider the conclusions of Theorems~ and~. Positive-characteristic extension conjecture. Theorems~ and~ hold over $k$ for $K$. The conjecture extends the preceding positive-characteristic results, which require $K$ to be both semisimple and cosemisimple; whether cosemisimplicity can be omitted remains open.